All these transforms are switching to an eigenbasis of some differential operator (that usually corresponds to a differential equation of interest). Spherical harmonics, Bessel and Henkel functions, which are the radial versions of sines/cosines and complex exponential, respectively, and on and on.
The next big jumps were to collections of functions not parameterized by subsets of R^n. Wavelets use a tree shapes parameter space.
There’s a whole, interesting area of overcomplete basis sets that I have been meaning to look into where you give up your basis functions being orthogonal and all those nice properties in exchange for having multiple options for adapting better to different signal characteristics.
I don’t think these transforms are going to be relevant to understanding neural nets, though. They are, by their nature, doing something with nonlinear structures in high dimensions which are not smoothly extended across their domain, which is the opposite problem all our current approaches to functional analysis deal with.
The next big jumps were to collections of functions not parameterized by subsets of R^n. Wavelets use a tree shapes parameter space.
There’s a whole, interesting area of overcomplete basis sets that I have been meaning to look into where you give up your basis functions being orthogonal and all those nice properties in exchange for having multiple options for adapting better to different signal characteristics.
I don’t think these transforms are going to be relevant to understanding neural nets, though. They are, by their nature, doing something with nonlinear structures in high dimensions which are not smoothly extended across their domain, which is the opposite problem all our current approaches to functional analysis deal with.